Damped Forced Vibration of Some Quasi-Linear Differential Systems

Abstract

There are many approaches for approximating solutions of nonlinear vibrating problems. The most common methods for constructing approximate analytical solutions to the nonlinear vibrating problems are the perturbation methods. These methods are developed to find only periodic vibrations of the nonlinear differential systems. In order to investigate the transients of nonlinear vibrations, Krylov and BogoLiubov introduced a perturbation method to discuss the transients in the second order autonomous systems with small nonlinearities. The method is well known as an "asymptotic averaging method" in the theory of nonlinear vibrations. Then the method was amplified and justified by BogoLiubov and Mitropolskii. These methods were applied to autonomous systems. Later, Arya and Bojadziev, Bojadziev and Hung, and Shamsul extended the Krylov-Bogo Liubov-Mitropolskii (KBM) method to sometime dependent nonlinear differential systems. In this dissertation, we extend the work of KBM and investigate some other time dependent non-linear differential systems. Firstly, a second order time dependent nonlinear differential system is considered. Then a new perturbation technique is developed to find an asymptotic solution of nonlinear vibrations in presence of a slowly decaying external force. We then find an asymptotic solution of a time dependent nonlinear differential system with slowly varying coefficients using the KBM method. Later, we find the perturbation solutions of damped forced vibrations using the modified KBM method, in which the coefficients change slowly varying with time. Further, this technique is used to obtain the second approximate solution of second order forced vibrations. Finally, this technique is used to obtain the higher approximate solution of an n-th order damped forced vibrating problem in the resonance case, and the stability of the stationary regime of vibrations has also been investigated. The methods are illustrated by several examples.